Finite Group Databasebeta

← C2×#21445026 #29655677

Subgroups — C2×#21445026 #29655677

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840112C211yesmaximal, characteristic, complemented
2C2×#16840047C211yesmaximal, characteristic, complemented
3C2×#16839967C211yesmaximal, characteristic
4#21445026C218yesmaximal, direct product factor
5C2×#16839511C211yesmaximal, characteristic, complemented
6C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
7C2×D4⋊#92418C211yesmaximal, characteristic
8C2×C61×#554C211yesmaximal, characteristic
9C22×#11758200—21no—
10C2×#14519777C2211yescharacteristic
11#16840112—22no—
12#16840112C2214yescomplemented
13#16840047—22no—
14#16840047C2214yescomplemented
15#16839967—22no—
16#16839967C2214yes—
17C2×#14519708—21no—
18C2×C4⋊#92418C2211yescharacteristic
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519701—22no—
21C2×#14519701—21no—
22C2×#14519700—21no—
23C2×C4×#92418C2211yescharacteristic
24#16839511C2214yescomplemented
25Q8⋊#92418C2214yescomplemented
26D4⋊#92418C2214yes—
27C61×#554—22no—
28C61×#554—22no—
29C61×#554—22no—
30C61×#554C2214yes—
31C2×C4⋊Q8×C61C2211yescharacteristic
32C2×#125×C61—21no—
33C2×C22⋊Q8×C61—21no—
34C2×C22⋊Q8×C61—22no—
35C2×C4×Q8×C61C2211yescharacteristic
36C2×C4×D4×C61—21no—
37C2×C4×D4×C61C2211yescharacteristic
38C22×C4⋊C4×C61—21no—
39C22×Dic122—21no—
40C2×#11758200—24no—
41C2×#11758200—44no—
42C22×#92418—21no—
43C22×#92418—21no—
44C22×#92418—21no—
45#14519777—21no—
46#14519777—21no—
47#14519777—21no—
48#14519777C2312yes—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51#14519708—24no—
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C4⋊#92418C2312yes—
55C4⋊#92418C2312yes—
56#14519701—24no—
57#14519701—28no—
58#14519700—24no—
59C4×#92418—21no—
60C4×#92418C2312yes—
61C22×Q8×C61D411yescharacteristic
62C22×Q8×C61D412yes—
63C22×D4×C61D411yescharacteristic
64C23×C4×C61—22no—
65C23×C4×C61—21no—
66C4⋊Q8×C61—21no—
67C4⋊Q8×C61—21no—
68C4⋊Q8×C61D412yes—
69C4⋊Q8×C61C2312yes—
70#125×C61—24no—
71#125×C61—24no—
72C22⋊Q8×C61—24no—
73C22⋊Q8×C61—28no—
74C22⋊Q8×C61—24no—
75C22⋊Q8×C61—28no—
76C4×Q8×C61—21no—
77C4×Q8×C61—21no—
78C4×Q8×C61D412yes—
79C4×Q8×C61C2312yes—
80C4×D4×C61—24no—
81C4×D4×C61—24no—
82C4×D4×C61—21no—
83C4×D4×C61—21no—
84C4×D4×C61D412yes—
85C4×D4×C61C2312yes—
86C2×C4⋊C4×C61—24no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—24no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61D411yescharacteristic
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61—21no—
97C2×C42×C61—21no—
98C2×C42×C61C2311yescharacteristic
99C24×C61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—44no—
105C2×Q8×C61C2×D412yes—
106C2×Q8×C61C2×D414yes—
107C2×D4×C61—21no—
108C2×D4×C61—22no—
109C2×D4×C61SD1614yes—
110C2×D4×C61C2×D412yes—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—28no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61—84no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×Dic61—41no—
125C2×Dic122—22no—
126C2×Dic122—42no—
127C2×Dic122—42no—
128C2×Dic122—41no—
129C4⋊C4×C61—28no—
130C4⋊C4×C61—28no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61C2×D412yes—
140C4⋊C4×C61C2×D412yes—
141C22⋊C4×C61—28no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C42×C61—22no—
145C42×C61—22no—
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61C2411yescharacteristic
150#11758200—48no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153C4⋊Dic61—41no—
154C4⋊Dic61—41no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—42no—
159C2×#92418—42no—
160C2×#92418—42no—
161C23×C61—22no—
162C23×C61—24no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61#12011yescharacteristic
166Q8×C61—22no—
167Q8×C61—22no—
168Q8×C61—44no—
169Q8×C61—42no—
170Q8×C61—82no—
171D4×C61—22no—
172D4×C61—24no—
173D4×C61C2×SD1614yes—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—24no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—44no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—44no—
186C2×C4×C61—88no—
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61#12011yescharacteristic
190C2×C4×C61C2×SD1612yes—
191C2×C4×C61#13512yes—
192C2×C4×C61#13712yes—
193C2×C4×C61#13712yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×Dic61—42no—
198C2×Dic61—82no—
199Dic122—42no—
200Dic122—42no—
201Dic122—82no—
202#92418—42no—
203#92418—42no—
204#92418—42no—
205C2×#1643—611nomaximal, Sylow
206C22×C61—24no—
207C22×C61—48no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61#44812yes—
211C22×C61#45212yes—
212C22×C61#45512yes—
213C22×C61C2×#12011yescharacteristic
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—42no—
217C4×C61—42no—
218C4×C61—42no—
219C4×C61—44no—
220C4×C61#44812yes—
221C4×C61#45212yes—
222C4×C61C22×SD1611yescharacteristic
223C4×C61C2×#13511yescharacteristic
224C4×C61C2×#13711yescharacteristic
225C4×C61C2×#13711yescharacteristic
226Dic61—82no—
227C2×#554D6111yescharacteristic
228C2×#487—611no—
229C2×#475—611no—
230C2×#461—611no—
231#1643—618no—
232C2×#333—611no—
233C2×Q8⋊C8—611no—
234C2×D4⋊C8—611no—
235C2×C61—44no—
236C2×C61#164314yescomplemented
237C2×C61C2×#44811yescharacteristic
238C2×C61C2×#45211yescharacteristic
239C2×C61C2×#45511yescharacteristic
240C22×SD16—1221no—
241#554—22no—
242#554—22no—
243#554—22no—
244#554C2×D6114yes—
245C2×C4⋊Q8—611no—
246C2×C4⋊Q8C2×D6111yescharacteristic
247C2×#125—21no—
248C2×C22⋊Q8—21no—
249C2×C22⋊Q8—22no—
250C2×C4×Q8C2×D6111yescharacteristic
251C2×C4×D4—21no—
252C2×C4×D4C2×D6111yescharacteristic
253C22×C4⋊C4—21no—
254#487—614no—
255#487—1222no—
256#475—614no—
257#475—1222no—
258#461—614no—
259#461—1222no—
260C2×#106—1221no—
261C2×C4⋊C8—611no—
262C2×C4⋊C8—611no—
263C2×#103—1221no—
264C2×#103—1222no—
265C2×#103—1221no—
266C2×C4×C8—611no—
267#333—614no—
268Q8⋊C8—614no—
269D4⋊C8—614no—
270C61C2×#164311yescharacteristic, complemented, Sylow
271C22×Q8—1221no—
272C22×Q8C22⋊D6111yescharacteristic
273C22×Q8C22⋊D6112yes—
274C22×D4C22⋊D6111yescharacteristic
275C23×C4—22no—
276C23×C4—21no—
277C2×SD16—1224no—
278C2×SD16—2444no—
279C22×C8—1221no—
280C22×C8—1221no—
281C22×C8—1221no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—612no—
285C4⋊Q8—1221no—
286C4⋊Q8—1221no—
287C4⋊Q8—1221no—
288C4⋊Q8C22⋊D6112yes—
289C4⋊Q8C22×D6112yes—
290#125—24no—
291#125—24no—
292C22⋊Q8—24no—
293C22⋊Q8—28no—
294C22⋊Q8—24no—
295C22⋊Q8—28no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8C22⋊D6112yes—
299C4×Q8C22×D6112yes—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4C22⋊D6112yes—
305C4×D4C22×D6112yes—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1221no—
315C2×C4⋊C4C22⋊D6111yescharacteristic
316C2×C4⋊C4C22⋊D6111yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C42—21no—
320C2×C42C22×D6111yescharacteristic
321#106—1224no—
322C4⋊C8—612no—
323C4⋊C8—612no—
324C4⋊C8—1221no—
325C4⋊C8—1221no—
326#103—1224no—
327#103—1228no—
328#103—1224no—
329C4×C8—612no—
330C4×C8—1221no—
331C24—21no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—44no—
337C2×Q8—1222no—
338C2×Q8—2442no—
339C2×Q8—2442no—
340C2×Q8—2441no—
341C2×Q8C2×C22⋊D6112yes—
342C2×Q8C2×C22⋊D6114yes—
343C2×D4—22no—
344C2×D4—21no—
345C2×D4#1175820114yes—
346C2×D4C2×C22⋊D6112yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—84no—
357C22×C4—2441no—
358C22×C4C2×C22⋊D6111yescharacteristic
359C22×C4C2×C22⋊D6111yescharacteristic
360C22×C4C2×C22⋊D6111yescharacteristic
361SD16—2448no—
362C2×C8—1222no—
363C2×C8—1222no—
364C2×C8—1222no—
365C2×C8—2442no—
366C2×C8—2442no—
367C2×C8—2442no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—1222no—
379C4⋊C4—1222no—
380C4⋊C4—2441no—
381C4⋊C4—2441no—
382C4⋊C4C2×C22⋊D6112yes—
383C4⋊C4C2×C22⋊D6112yes—
384C22⋊C4—28no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C42—22no—
388C42—22no—
389C42C2×C22⋊D6111yescharacteristic
390C42C2×C22⋊D6111yescharacteristic
391C42C2×C22⋊D6111yescharacteristic
392C42C23×D6111yescharacteristic
393C23—22no—
394C23—24no—
395C23—42no—
396C23—42no—
397C23#1451974611yescharacteristic
398Q8—22no—
399Q8—22no—
400Q8—44no—
401Q8—42no—
402Q8—82no—
403Q8—4882no—
404Q8—2442no—
405Q8—2442no—
406D4—22no—
407D4—24no—
408D4C2×#1175820114yes—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—88no—
422C2×C4—4882no—
423C2×C4—2442no—
424C2×C4#1451974611yescharacteristic
425C2×C4#1451974611yescharacteristic
426C2×C4#1451974611yescharacteristic
427C2×C4C2×#1175820112yes—
428C2×C4#1451979712yes—
429C2×C4#1451980712yes—
430C2×C4#1451980812yes—
431C2×C4C22×C22⋊D6111yescharacteristic
432C2×C4C22×C22⋊D6111yescharacteristic
433C2×C4C22×C22⋊D6111yescharacteristic
434C8—2442no—
435C8—2442no—
436C8—2442no—
437C22—24no—
438C22—42no—
439C22—48no—
440C22—42no—
441C22#1683990112yes—
442C22#1683992412yes—
443C22#1683994012yes—
444C22C2×#1451974611yescharacteristic
445C4—21no—
446C4—21no—
447C4—42no—
448C4—44no—
449C4—42no—
450C4—42no—
451C4—4882no—
452C4#1683990112yes—
453C4#1683992412yes—
454C4C22×#1175820111yescharacteristic
455C4C2×#1451979711yescharacteristic
456C4C2×#1451980711yescharacteristic
457C4C2×#1451980811yescharacteristic
458C2—44no—
459C2#2144502614yesdirect product factor
460C2C2×#1683990111yescharacteristic
461C2C2×#1683992411yescharacteristic
462C2C2×#1683994011yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#554C2
socleC23×C61#120
prime radicalC22×D4×C61D4
hypercenterC2×#554D61
hypocenterC61C2×#1643
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#554

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840112C2semidirect
2. C2×#16840047C2semidirect
4. #21445026C2direct (N × H)
5. C2×#16839511C2semidirect
6. C2×Q8⋊#92418C2semidirect
12. #16840112C22semidirect
14. #16840047C22semidirect
24. #16839511C22semidirect
25. Q8⋊#92418C22semidirect
236. C2×C61#1643semidirect
270. C61C2×#1643semidirect
459. C2#21445026direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168401122
C2×#168400472
C2×#168399672
#214450262
C2×#168395112
C2×Q8⋊#924182
C2×D4⋊#924182
C2×C61×#5542
C2×#164361